Klein bottles and nuclear fusion

You may have created a Möbius strip in school: a strip of paper, twisted and glued to itself. From this simple procedure emerges the paradox of an object with only one side.

In 1882, the year of Emmy Noether’s birth, the great Felix Klein invented what everyone now calls the “Klein bottle”. This is an extension, to three dimensions, of the 2D surface of the Möbius strip. (Well, sort of. See the link for a more complete description.) They’re much harder to make, but not impossible.

The Klein bottle is usually thought of as belonging to the realm of pure and abstract, and perhaps recreational, mathematics: a mathematical curiosity, instructive and interesting, but probably without much application.

Recently C. B. Smiet showed that Klein bottles may indeed have practical applications to the analysis of fusion plasmas in toroidal confinement vessels. He begins with this charming paragraph, unusual for a paper in a mainstream physics journal:

This paper does not solve any key problems on the path to realizing fusion, nor does it claim that its conclusions are important. It exists purely because of the esthetically pleasing nature of the novel observations it contains, highlighting profound connections between pure mathematics, applied mathematics, and physics.

The author then goes on to surprise us by delving deeply into experimental data and showing that Klein bottles may nevertheless have something to say about magnetic field configurations in real devices. The mathematics provides insight into how to avoid a class of instabilities in these devices.

I don’t want to give the impression that I think there’s any chance of fusion becoming a commercially relevant energy source. Apart from that, this paper is a fascinating example of the eternally surprising connections between abstract reasoning in mathematics and the complex and contingent world around us.